Let $$f(x) = (...((x - 2)^2 - 2)^2 - 2)^2... - 2)^2$$ (here there are $n$ brackets $( )$). Find $f''(0)$

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

For all positive real numbers $a,b,c$, prove that $$\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)$$

The sum $$\sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1}$$ can be expressed as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find the remainder when $a+b$ is divided by $1000.$

Show that there exists a sequence $x_1,x_2,...$ of natural numbers in which every natural number occurs exactly once, such that the sums $\sum_{i=1}^n \frac{1}{x_i}$, $n = 1,2,3,...$, include all natural numbers.

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]

Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root? $\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$

Froa nay real $x$, we denote $[x]$, the integer part of $x$ and with $\{x\}$ the fractional part of $x$, such that $x=[x]+\{x\}$. a) Find at least one real $x$ such that$\{x\}+\left\{\frac{1}{x}\right\}=1$ b) Find all rationals $x$ such that $\{x\}+\left\{\frac{1}{x}\right\}=1$

[b]a)[/b] Prove that the function $ f:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} , $ given as $ f(n)=n+(-1)^n $ is bijective. [b]b)[/b] Find all surjective functions $ g:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} $ that have the property that $ g(n)\ge n+(-1)^n , $ for any nonnegative integer.

Consider sequences $a$ of the form $a = (a_1, a_2, ... , a_{20})$ such that each term $a_i$ is either $0$ or $1$. For each such sequence $a$, we can produce a sequence $b = (b_1, b_2, ..., b_{20})$, where $$b_i\begin{cases} a_i + a_{i+1} & i = 1 \\ a_{i-1} + a_i + a_{i+1} & 1 < i < 20\\ a_{i-1} + a_i &i = 20 \end{cases}$$

Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.

The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$, \[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \] Find $a_{2004}$.

[u]Part 6 [/u] [b]p16.[/b] Le[b][/b]t $p(x)$ and $q(x)$ be polynomials with integer coefficients satisfying $p(1) = q(1)$. Find the greatest integer $n$ such that $\frac{p(2023)-q(2023)}{n}$ is an integer no matter what $p(x)$ and $q(x)$ are. [b]p17.[/b] Find all ordered pairs of integers $(m,n)$ that satisfy $n^3 +m^3 +231 = n^2m^2 +nm.$ [b]p18.[/b] Ben rolls the frustum-shaped piece of candy (shown below) in such a way that the lateral area is always in contact with the table. He rolls the candy until it returns to its original position and orientation. Given that $AB = 4$ and $BD =CD = 3$, find the length of the path traced by $A$. [u]Part 7 [/u] [b]p19.[/b] In their science class, Adam, Chris, Eddie and Sam are independently and randomly assigned an integer grade between $70$ and $79$ inclusive. Given that they each have a distinct grade, what is the expected value of the maximum grade among their four grades? [b]p20.[/b] Let $ABCD$ be a regular tetrahedron with side length $2$. Let point $E$ be the foot of the perpendicular from $D$ to the plane containing $\vartriangle ABC$. There exist two distinct spheres $\omega_1$ and $\omega_2$, centered at points $O_1$ and $O_2$ respectively, such that both $O_1$ and $O_2$ lie on $\overrightarrow{DE}$ and both spheres are tangent to all four of the planes $ABC$, $BCD$, $CDA$, and $DAB$. Find the sum of the volumes of $\omega_1$ and $\omega_2$. [b]p21.[/b] Evaluate $$\sum^{\infty}_{i=0}\sum^{\infty}_{j=0}\sum^{\infty}_{k=0} \frac{1}{(i + j +k +1)2^{i+j+k+1}}.$$ [u]Part 8 [/u] [b]p22.[/b] In $\vartriangle ABC$, let $I_A$, $I_B$ , and $I_C$ denote the $A$, $B$, and $C$-excenters, respectively. Given that $AB = 15$, $BC = 14$ and $C A = 13$, find $\frac{[I_A I_B I_C ]}{[ABC]}$ . [b]p23.[/b] The polynomial $x +2x^2 +3x^3 +4x^4 +5x^5 +6x^6 +5x^7 +4x^8 +3x^9 +2x^{10} +x^{11}$ has distinct complex roots $z_1, z_2, ..., z_n$. Find $$\sum^n_{k=1} |R(z^2n))|+|I(z^2n)|,$$ where $R(z)$ and $I(z)$ indicate the real and imaginary parts of $z$, respectively. Express your answer in simplest radical form. [b]p24.[/b] Given that $\sin 33^o +2\sin 161^o \cdot \sin 38^o = \sin n^o$ , compute the least positive integer value of $n$. [u]Part 9[/u] [b]p25.[/b] Submit a prime between $2$ and $2023$, inclusive. If you don’t, or if you submit the same number as another team’s submission, you will receive $0$ points. Otherwise, your score will be $\min \left(30, \lfloor 4 \cdot ln(x) \rfloor \right)$, where $x$ is the positive difference between your submission and the closest valid submission made by another team. [b]p26.[/b] Sam, Derek, Jacob, andMuztaba are eating a very large pizza with $2023$ slices. Due to dietary preferences, Sam will only eat an even number of slices, Derek will only eat a multiple of $3$ slices, Jacob will only eat a multiple of $5$ slices, andMuztaba will only eat a multiple of $7$ slices. How many ways are there for Sam, Derek, Jacob, andMuztaba to eat the pizza, given that all slices are identical and order of slices eaten is irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be: irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be: $$\max \left( 0, \left\lfloor 30 \left( 1-2\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$ [b]p27.[/b] Let $ \Omega_(k)$ denote the number of perfect square divisors of $k$. Compute $$\sum^{10000}_{k=1} \Omega_(k).$$ If your answer is $A$ and the correct answer is $C$, the number of points you recieve will be $$\max \left( 0, \left\lfloor 30 \left( 1-4\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$ PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3267911p30056982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.

The numbers $16$ and $25$ are a pair of consecutive perfect squares whose difference is $9$. How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$? $\textbf{(A) } 674 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1010 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2017$

For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers. Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one. R. Salimov

Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $$(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0$$ then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one.

Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.

Let $X$ be a set with $|X| = n$ , and let $X_1 , X_2 ,... X_n$ be the $n$subsets eith $|X_j| \geq 2$, for $1 \leq j \leq n$. Suppose for each $2$ element subset $Y$ of $X$, there is a unique $j$ in the set $1,2,3....,n$ such that $Y \subset X_j$ . Prove that $X_j \cap X_k \not= \Phi$ for all $1 \leq j < k \leq n$

Consider an infinite sequence of reals $x_1, x_2, x_3, ...$ such that $x_1 = 1$, $x_2 =\frac{2\sqrt3}{3}$ and with the recursive relationship $$n^2 (x_n - x_{n-1} - x_{n-2}) - n(3x_n + 2x_{n-1} + x_{n-2}) + (x_nx_{n-1}x_{n-2} + 2x_n) = 0.$$ Find $x_{2019}$.

Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.

Danielle writes a sign '+' or '-' in each of the next $64$ spaces: $$\_\_1 \_\_2 \_\_3 \_\_4 \text{ }.... \text{ }\_\_63 \_\_64=2024$$ such that the equality holds. Find the largest number of negative signs Danielle can use.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(2m+2n)=f(m)f(n)$ for all natural numbers $m,n$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$

A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition: \[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\] Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds: \[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]